Chaos is widely understood as being a consequence of sensitive dependenceupon initial conditions. Despite their overall intrinsic instability,trajectories may be very stronglyconvergent in phase space over extremely long periods, as revealed by ourinvestigation of a simple chaotic system (a realistic model for smallbodies in a turbulent flow).We establish that this strong convergence is a multi-facettedphenomenon, in which theclustering is intense, widespread and balanced by lacunarity of other regions.Power laws, indicative of scale-free features, characterise the distribution ofparticles in the system. We use large-deviation and extreme-value statistics toexplain the effect. Our results show that the interpretation of the 'butterflyeffect' needs to be carefully qualified. This notion of convergent chaos, whichimplies the existence of conditions for which uncertainties areunexpectedly small, mayalso be relevant to the valuation of insurance and futures contracts.